Power Rule of derivative is an essential formula in differential calculus. Now we shall prove this formula by definition or first principle.
Let us suppose that the function of the form , where is any constant power.
First we take the increment or small change in the function.
Putting the value of function in the above equation, we get
Taking common from the above equation, we get
Now taking common , we get
Expanding the above expression using binomial series, we get
Dividing both sides by , we get
Taking limit of both sides as , we have
This shows that the derivative of power is .
Example: Find the derivative of
We have the given function as
Differentiation with respect to variable , we get
Now using the power rule and constant function rule, we have